Number and shape patterns Knowledge-Based Study

Study notes

1. What Is a Pattern?

   A pattern is an ordered arrangement of numbers, shapes, or objects that follows a specific, consistent mathematical rule. Each number in a sequence is called a term.

   Term-to-Term Rules

   A term-to-term rule explains how to get from one term to the immediate next term in a sequence.

   • Arithmetic (Linear) Patterns: Formed by adding or subtracting a constant value called the common difference. For example, in the sequence 4, 7, 10, 13, ..., the rule is "add 3".
   • Geometric Patterns: Formed by multiplying or dividing by a constant factor. For example, in the sequence 2, 6, 18, 54, ..., the rule is "multiply by 3".

   To determine the term-to-term rule of an arithmetic sequence, subtract any term from the term that immediately follows it: $d = a_{n} - a_{n-1}$.

   Visualising Patterns with Shapes

   Geometric or shape patterns use visual figures that grow or change from one stage to the next. We can translate any growing shape pattern into a numerical sequence by counting key features, such as line segments, vertices, or unit tiles.

   Step-by-Step Translation

   1. Identify the variable feature: Determine what is changing across consecutive figures (for example, the total number of matchsticks or square tiles).
   2. Record values in a table: Create a table of values linking the position number (or stage $n$) to the count of parts.
   3. Analyse the differences: Look at the difference between consecutive values to establish the rate of growth.

   For instance, if Stage 1 has 4 matchsticks forming 1 square, Stage 2 has 7 matchsticks forming 2 connected squares, and Stage 3 has 10 matchsticks forming 3 connected squares, the numerical sequence is 4, 7, 10, ... with a common difference of 3.

   Why Do We Need a General Rule?

   A term-to-term rule requires calculating every single step in order. If we want to find the 50th or 100th term, we need a position-to-term rule, often expressed as an algebraic formula using the variable $n$, where $n$ represents the position number.

   Finding the Rule for Linear Patterns

   For any sequence with a constant difference $d$, the $n$th term has the general form:

   $$\text{Term} = d \cdot n + c$$

   To find the constant $c$:

   • Multiply the position number $n = 1$ by the common difference $d$.
   • Compare the result $d \cdot 1$ to the actual first term $a_1$.
   • Set $c = a_1 - d$.

   Using our matchstick example (4, 7, 10, ...):

   • Common difference: $d = 3$.
   • Trial term: $3n$. For $n = 1$, $3(1) = 3$.
   • To reach the first term 4, add 1.
   • Therefore, the $n$th term rule is $3n + 1$.

   Predicting Distant Terms

   Once the general rule is established, you can find any term without drawing or listing all preceding steps. To find the 25th term of the pattern defined by $3n + 1$:

   $$\text{Term}_{25} = 3(25) + 1 = 75 + 1 = 76$$

   Determining Whether a Value Belongs to a Pattern

   To check if a specific number belongs to a sequence, set the general expression equal to that number and solve for $n$. If $n$ is a positive integer, the number is part of the sequence.

   Example: Is 100 a term in the sequence $3n + 1$?

   $$3n + 1 = 100$$

   $$3n = 99$$

   $$n = 33$$

   Because 33 is a whole number, 100 is indeed the 33rd term of the pattern.